.. _use-case-branin: The Branin test case ==================== Introduction ------------ The Branin function is defined in 2 dimensions based on the functions :math:g: .. math:: g(u_1, u_2) = \frac{\left(u_2-5.1\frac{u_1^2}{4\pi^2}+5\frac{u_1}{\pi}-6\right)^2+10\left(1-\frac{1}{8 \pi}\right) \cos(u_1)+10-54.8104}{51.9496} and :math:t: .. math:: t(x_1, x2) = (15 x_1 - 5, 15 x_2)^T. Finally, the Branin function is the composition of the two previous functions: .. math:: f_{Branin}(x_1, x_2) = g \circ t(x_1, x_2) for any :math:\mathbf{x} \in [0, 1]^2. There are three global minimas: .. math:: \mathbf{x}^\star=(0.123895, 0.818329), .. math:: \mathbf{x}^\star=(0.542773, 0.151666), and : .. math:: \mathbf{x}^\star=(0.961652, 0.165000) where the function value is: .. math:: f_{min} = f_{Branin}(\mathbf{x}^\star) = -0.97947643837. We assume that the output of the Branin function is noisy, with a gaussian noise. In other words, the objective function is: .. math:: f(x_1, x_2) = f_{Branin}(x_1, x_2) + \epsilon where :math:\epsilon is a random variable with gaussian distribution. This time the AEI formulation is used, meaning that the objective has two outputs: the first one is the objective function value and the second one is the noise variance. Here we assume a constant noise variance: .. math:: \sigma_{\epsilon} = 0.1. References ---------- * Dixon, L. C. W., & Szego, G. P. (1978). The global optimization problem: an introduction. Towards global optimization, 2, 1-15. * Forrester, A., Sobester, A., & Keane, A. (2008). Engineering design via surrogate modelling: a practical guide. Wiley. * Global Optimization Test Problems. Retrieved June 2013, from http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO.htm. * Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf. * Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization. Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import branin_function as branin_function >>> # Load the Branin-Hoo test case >>> bm = branin_function.BraninModel() API documentation ----------------- See :class:~openturns.usecases.branin_function.BraninModel. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_numerical_methods/optimization/images/thumb/sphx_glr_plot_ego_thumb.png :alt: :ref:sphx_glr_auto_numerical_methods_optimization_plot_ego.py .. raw:: html
.. toctree:: :hidden: /auto_numerical_methods/optimization/plot_ego